The Minnesota prior, also called the Litterman prior, is a shrinkage prior for autoregressive parameters in vector autoregressive (VAR) models. There are many variations of the Minnesota prior. This Minnesota prior is calculated as presented in Lutkepohl (2005, p. 225), and returns one or more prior covariance matrices in an array.

`MinnesotaPrior(J, lags=c(1,2), lambda=1, theta=0.5, sigma)`

J

This is the scalar number of time-series in the VAR.

lags

This accepts an integer vector of lags of the autoregressive parameters. The lags are not required to be successive.

lambda

This accepts a scalar, positive-only hyperparameter that controls how tightly the parameter of the first lag is concentrated around zero. A smaller value results in smaller diagonal variance. When equal to zero, the posterior equals the prior and data is not influential. When equal to infinity, no shrinkage occurs and posterior expectations are closest to estimates from ordinary least squares (OLS). It has been asserted that as the number, \(J\), of time-series increases, this hyperparameter should decrease.

theta

This accepts a scalar hyperparameter in the interval [0,1]. When one, off-diagonal elements have variance similar or equal to diagonal elements. When zero, off-diagonal elements have zero variance. A smaller value is associated with less off-diagonal variance.

sigma

This accepts a vector of length \(J\) of residual standard deviations of the dependent variables given the expectations.

This function returns a \(J \times J \times L\) array for \(J\) time-series and \(L\) lags.

The Minnesota prior was introduced in Doan, Litterman, and Sims (1984) as a shrinkage prior for autoregressive parameters in vector autoregressive (VAR) models. The Minnesota prior was reviewed in Litterman (1986), and numerous variations have been presented since. This is the version of the Minnesota prior as described in Lutkepohl (2005, p. 225) for stationary time-series.

Given one or more \(J \times J\) matrices of autoregressive
parameters in a VAR model, the user specifies two tuning
hyperparameters for the Minnesota prior: `lambda`

and
`theta`

. Each iteration of the numerical approximation algorithm,
the latest vector of residual standard deviation parameters is
supplied to the `MinnesotaPrior`

function, which then returns an
array that contains one or more prior covariance matrices for the
autoregressive parameters. Multiple prior covariance matrices are
returned when multiple lags are specified. The tuning hyperparameters,
`lambda`

and `theta`

, can be estimated from the data via
hierarchical Bayes.

It is important to note that the Minnesota prior does not technically
return a covariance matrix, because the matrix is not symmetric, and
therefore not positive-definite. For this reason, a Minnesota prior
covariance matrix should not be supplied as a covariance matrix to a
multivariate normal distribution, such as with the `dmvn`

function, though it would be accepted and then (incorrectly)
converted to a symmetric matrix. Instead, `dnormv`

should
be used for element-wise evaluation.

While the Minnesota prior is used to specify the prior covariance for VAR autoregressive parameters, prior means are often all set to zero, or sometimes the first lag is set to an identity matrix.

An example is provided in the Examples vignette.

Doan, T., Litterman, R.B. and Sims, C.A. (1984). "Forecasting and
Conditional Projection using Realistic Prior Distributions".
*Econometric Reviews*, 3, p. 1--144.

Litterman, R.B. (1986). "Forecasting with Bayesian Vector
Autoregressions - Five Years of Experience". *Journal of
Business & Economic Statistics*, 4, p. 25--38.

Lutkepohl, H. (2005). "New Introduction to Multiple Time Series Analysis". Springer, Germany.

`dmvn`

,
`dnormv`

, and
`LaplacesDemon`

.